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The Continuous Time Random Walk (CTRW) provides a mathematical framework for the study of transport in heterogenous media. It is much more general than usual transport models based on (local, Markovian) Ordinary Differential Equations, and in particular can handle transport in systems without characteristic scales (such as systems in a state of [[Self-Organised Criticality]] or SOC).
The Continuous Time Random Walk (CTRW) provides a mathematical framework for the study of transport in heterogenous media. It is much more general than usual transport models based on (local, Markovian) Ordinary Differential Equations, and in particular can handle transport in systems without characteristic scales (such as systems in a state of [[Self-Organised Criticality]] or SOC).


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CTRW theory
CTRW theory
<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>
&lt;ref&gt;R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307&lt;/ref&gt;
starts from the single-particle step distribution function (in one dimension)
starts from the single-particle step distribution function (in one dimension)


:<math>\xi(\Delta x, \Delta t; x, t)</math>
:&lt;math&gt;\xi(\Delta x, \Delta t; x, t)&lt;/math&gt;


giving the probability that a particle, located at position ''x'' at time ''t'', takes a step of size ''&Delta;x'' after waiting a time ''&Delta;t'' &gt; 0. All particles are assumed to be identical.
giving the probability that a particle, located at position ''x'' at time ''t'', takes a step of size ''&amp;Delta;x'' after waiting a time ''&amp;Delta;t'' &amp;gt; 0. All particles are assumed to be identical.


== The Master Equation ==
== The Master Equation ==
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This equation is a (Generalized) Master Equation.
This equation is a (Generalized) Master Equation.


In standard CTRW theory, it is customary to assume that the single particle step distribution is ''separable'', i.e., that ''&Delta;x'' is independent from ''&Delta;t'', so that
In standard CTRW theory, it is customary to assume that the single particle step distribution is ''separable'', i.e., that ''&amp;Delta;x'' is independent from ''&amp;Delta;t'', so that


: <math>\xi( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \psi(\Delta t; x,t)</math>
: &lt;math&gt;\xi( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \psi(\Delta t; x,t)&lt;/math&gt;


In addition, homogeneity in space and time is assumed (i.e., ''p'' and ''&psi;'' do not depend on ''x'' and ''t''). However, recently it was shown that a Master Equation can also be derived in the case that ''p'' depends on ''x'' and ''t'', while ''&psi;'' depends on ''x'' (but not ''t'').
In addition, homogeneity in space and time is assumed (i.e., ''p'' and ''&amp;psi;'' do not depend on ''x'' and ''t''). However, recently it was shown that a Master Equation can also be derived in the case that ''p'' depends on ''x'' and ''t'', while ''&amp;psi;'' depends on ''x'' (but not ''t'').
<ref>[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, ''Probabilistic finite-size transport models for fusion: anomalous transport and scaling laws'', Phys. Plasmas '''11''', 5 (2004) 2272]</ref>
&lt;ref&gt;[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, ''Probabilistic finite-size transport models for fusion: anomalous transport and scaling laws'', Phys. Plasmas '''11''', 5 (2004) 2272]&lt;/ref&gt;
This significant extension of the standard CTRW model has led to the development of a model with very interesting properties from the point of view of plasma transport (see the cited reference and <ref>[http://link.aip.org/link/?PHPAEN/11/3787/1 B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]</ref>).
This significant extension of the standard CTRW model has led to the development of a model with very interesting properties from the point of view of plasma transport (see the cited reference and &lt;ref&gt;[http://link.aip.org/link/?PHPAEN/11/3787/1 B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]&lt;/ref&gt;).


The Generalized Master Equation (GME) can be written in the form
The Generalized Master Equation (GME) can be written in the form


:<math>\frac{\partial n(x,t)}{\partial t} = \int_0^t \left ( \int{K(x-x',t-t',x',t')n(x',t')dx'} - n(x,t')\int{K(x-x',t-t',x',t')dx'}\right )dt'</math>
:&lt;math&gt;\frac{\partial n(x,t)}{\partial t} = \int_0^t \left ( \int{K(x-x',t-t',x',t')n(x',t')dx'} - n(x,t')\int{K(x-x',t-t',x',t')dx'}\right )dt'&lt;/math&gt;


where ''n'' is the particle (probability) density, and ''K'' a kernel of the form
where ''n'' is the particle (probability) density, and ''K'' a kernel of the form


:<math>K( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \phi(\Delta t; x)</math>
:&lt;math&gt;K( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \phi(\Delta t; x)&lt;/math&gt;


The GME is an integro-differential equation, generalizing the usual (partial differential) equations for transport.  
The GME is an integro-differential equation, generalizing the usual (partial differential) equations for transport.  
The particle flux at any point in space depends on the global distribution of the transported particle density field, and on its history (although the history dependence can be eliminated by choosing a Markovian waiting time distribution).
The particle flux at any point in space depends on the global distribution of the transported particle density field, and on its history (although the history dependence can be eliminated by choosing a Markovian waiting time distribution).


The treatment of boundary conditions in a GME is different from standard differential equations. <ref>[http://dx.doi.org/10.1088/1751-8113/41/21/215004 B.Ph. van Milligen, I. Calvo, and R. Sánchez, ''Continuous time random walks in finite domains and general boundary conditions: some formal considerations'', J. Phys. A: Math. Theor. '''41''' (2008) 215004]</ref>
The treatment of boundary conditions in a GME is different from standard differential equations. &lt;ref&gt;[http://dx.doi.org/10.1088/1751-8113/41/21/215004 B.Ph. van Milligen, I. Calvo, and R. Sánchez, ''Continuous time random walks in finite domains and general boundary conditions: some formal considerations'', J. Phys. A: Math. Theor. '''41''' (2008) 215004]&lt;/ref&gt;
The final (quasi) steady state of the system is a function of the balance between sources and sinks, rather than of imposed values or gradients at the system boundaries.
The final (quasi) steady state of the system is a function of the balance between sources and sinks, rather than of imposed values or gradients at the system boundaries.


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While the Master Equation corresponds to a CTRW in the limit of many particles, the  
While the Master Equation corresponds to a CTRW in the limit of many particles, the  
Fractional Differential Equation (FDE) corresponds to a Master Equation in the ''fluid limit''.
Fractional Differential Equation (FDE) corresponds to a Master Equation in the ''fluid limit''.
<ref>[http://link.aps.org/doi/10.1103/PhysRevE.71.011111 R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, ''Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations'', Phys. Rev. E '''71''' (2005) 011111]</ref>
&lt;ref&gt;[http://link.aps.org/doi/10.1103/PhysRevE.71.011111 R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, ''Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations'', Phys. Rev. E '''71''' (2005) 011111]&lt;/ref&gt;
The fluid limit is the limit in which only the part of the dynamics that is dominant for large scales and long times is retained, and is useful for understanding the steady state properties of a solution.
The fluid limit is the limit in which only the part of the dynamics that is dominant for large scales and long times is retained, and is useful for understanding the steady state properties of a solution.


To obtain the fractional differential operators, it is necessary to make an assumption regarding the shape of the distributions appearing in the kernel ''K''. Invoking the Generalized Limit Theorem for the sums of random variables,
To obtain the fractional differential operators, it is necessary to make an assumption regarding the shape of the distributions appearing in the kernel ''K''. Invoking the Generalized Limit Theorem for the sums of random variables,
<ref>B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)</ref>
&lt;ref&gt;B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)&lt;/ref&gt;
these distributions are taken to be [[:Wikipedia:Stable_distribution|Lévy distributions]].  
these distributions are taken to be [[:Wikipedia:Stable_distribution|Lévy distributions]].  
While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since &Delta;''t'' &gt; 0.
While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since &amp;Delta;''t'' &amp;gt; 0.
This choice allows modelling both  
This choice allows modelling both  
[[Non-diffusive transport|sub- and super-diffusive transport]], and in the appropriate limit, standard ("Fickian") transport is recovered.
[[Non-diffusive transport|sub- and super-diffusive transport]], and in the appropriate limit, standard (&quot;Fickian&quot;) transport is recovered.
If nothing else, this serves to show that all of the above constitute generalizations (on various levels) of the usual transport equations.
If nothing else, this serves to show that all of the above constitute generalizations (on various levels) of the usual transport equations.


The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,  
The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,  
<ref>[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>
&lt;ref&gt;[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]&lt;/ref&gt;
whereas the GME must be iterated in time.  
whereas the GME must be iterated in time.  
The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems.
The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems.
<ref>[http://link.aip.org/link/?PHPAEN/13/082308/1 D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]</ref>
&lt;ref&gt;[http://link.aip.org/link/?PHPAEN/13/082308/1 D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]&lt;/ref&gt;
On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.
On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.
<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>
&lt;ref&gt;[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]&lt;/ref&gt;


== References ==
== References ==
<references />
&lt;references /&gt;
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